THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Probing quantum and classical noise in nano circuits ARSALAN POURKABIRIAN Department of Microtechnology and Nanoscience CHALMERS UNIVERSITY OF TECHNOLOGY Goteborg,¨ Sweden 2014 Probing quantum and classical noise in nano circuits ARSALAN POURKABIRIAN ISBN 978-91-7597-117-9 ©ARSALAN POURKABIRIAN, 2014 Doktorsavhandlingar vid Chalmers tekniska hogskola¨ Ny serie nr 3798 ISSN 1652-0769 Chalmers University of Technology Department of Microtechnology and Nanoscience Quantum Device Physics Laboratory Experimental Mesoscopic Physics Group SE-412 96 Goteborg,¨ Sweden Telephone: +46 (0)31 - 772 1000 ISSN 1652-0769 Technical Report MC2-293 Chalmers Reproservice Goteborg,¨ Sweden 2014 iii Probing quantum and classical noise in nano circuits ARSALAN POURKABIRIAN Department of Microtechnology and Nanoscience Chalmers University of Technology, 2014 Abstract This thesis presents measurements of classical and quantum noise in nano circuits. The first part of the thesis, covers extensive measurements on charge noise sources. Low-frequency charge noise with the power spectrum close to 1/f (where f is the frequency) has been observed in a variety of systems. Despite the large theoretical and experimental efforts during the past three decades, the origin of this noise is still unknown. One of the best platforms to study this noise is the single electron transistor (SET) which is extremely sensitive to charge. We have exploited this unique charge sensitivity to probe the charge noise sources. We have measured the temperature and the bias dependence of the charge noise and concluded that the two-level fluctuators (TLFs) which cause the charge noise have a temperature which is closer to the temperature of the electrons on the SET rather than to the temperature of the phonos underneath the SET. This suggests that most probably the charge noise sources are in the vicinity of the SET and can ther- malize with SET electrons through quantum tunneling which limits their distribution to within a few nanometers around the SET. In another set of measurements, we have probed the TLFs when they are pushed out of equilibrium by an external electric field. The relaxation process of the TLFs causes a charge drift which we have measured using a SET over four decades of time. We found that this drift is logarithmic in time and by comparing it to theory we could extract the density of TLFs. Studying how the drift depends on temperature and electric field, we can also conclude that the switching of the TLFs is due to quantum tunneling and not due to thermal activation. The second part of the thesis covers experiments related to vacuum fluctuations. Vacuum fluctuations are one of the most interesting predictions of the quantum me- chanics. We have demonstrated the first observation of the dynamical Casimir effect, which is generation of real photons out of the vacuum by modulation of a mirror at relativistic speeds. We show broad band generation of photons and demonstrate two-mode squeezing of this radiation. In another experiment, we have measured the strength of these vacuum fluctuations by using an artificial atom in front of a mirror as our quantum probe. In the last part of the thesis, we present preliminary results for characterization of the system consisting of two artificial atoms in front of a mirror, a system which can potentially exploited for studying the interaction of artificial atoms through exchange of photons. Keywords: charge noise, two-level fluctuator, SET, SQUID, transmon, vacuum fluc- tuations, circuit-QED, dynamical Casimir effect, artificial atom. iv v List of Publications This thesis is based on the work contained in the following papers: I: Thermal properties of charge noise sources. M.V. Gustafsson∗, A. Pourkabirian∗, G. Johansson, J. Clarke and P. Delsing. Physical Review B 88, 245410, 2013. (* Joint first authorship) II: Non-equilibrium probing of two-level charge fluctuators using the step re- sponse of a single electron transistor. A. Pourkabirian, M.V. Gustafsson, G. Johansson, J. Clarke and P. Delsing. In review (arXiv:1408.6496). Non-equilibrium probing of two-level charge fluctuators using the step response of a single electron transistor: Supplemental Material. III: Observation of the dynamical Casimir effect in a superconducting circuit. C.M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J.R. Johansson, T. Duty, F. Nori and P. Delsing. Nature 479, 376, 2011. Observation of the dynamical Casimir effect in a superconducting circuit: Sup- plementary Information. IV: Probing the quantum vacuum with an atom in front of a mirror. I.-C Hoi, A.F. Kockum, L. Tornberg, A. Pourkabirian, G. Johansson, P. Delsing and C.M. Wilson. In review (arXiv:1410.8840). V: Scattering properties of two artificial atoms in front of a mirror. A. Pourkabirian, S.R. Sathyamoorthy, G. Johansson, C.M. Wilson and P. Delsing. Manuscript. vi Contents 1 Introduction 1 2 Theoretical Background 5 2.1 Single electronics . .5 2.2 Single electron box (SEB) . .6 2.3 Single electron transistor (SET) . .8 2.4 Noise in SETs . .9 2.4.1 Random fluctuations and noise . .9 2.4.2 Thermal noise . .9 2.4.3 Shot noise . 10 2.4.4 Low-frequency charge noise . 10 2.5 Ensembles of two-level fluctuators . 11 2.5.1 Microscopic models for two-level fluctuators . 12 2.5.2 Thermal properties of TLFs . 12 2.5.3 TLFs in an external electric field . 13 2.5.4 Step response; single TLF . 13 2.5.5 Step response; ensemble of TLFs . 15 2.6 Circuit quantum electrodynamics (circuit QED) . 17 2.6.1 Josephson junctions and SQUID . 18 2.6.2 Artificial atoms . 20 2.6.3 Single Cooper pair box (SCB) . 20 2.6.4 Decoherence in a qubit . 22 2.6.5 Transmon . 23 2.6.6 Atom-light interaction . 23 2.7 Vacuum fluctuations . 26 2.8 Dynamical Casimir effect . 26 3 Experimental Techniques 33 3.1 Fabrication . 33 3.2 Cryogenics . 36 3.2.1 Low-frequency measurements setup . 39 3.2.2 Microwave reflectometry setup . 39 vii viii CONTENTS 4 Results: Probing Two-Level Fluctuators 45 4.1 Equilibrium properties of TLFs: Temperature dependence of noise . 45 4.2 Non-equilibrium probing of TLFs: Step response . 51 4.3 Conclusions and outlook . 57 5 Results: Probing Vacuum Fluctuations 59 5.1 Observation of the dynamical Casimir effect . 59 5.2 Probing the vacuum fluctuations with an artificial atom in front of a mirror . 64 5.3 Scattering properties of two atoms in front of a mirror . 69 Appendix I 74 Acknowledgements 78 Bibliography 81 Appended Papers 93 Paper I 95 Paper II 104 Paper II: Supplemental Material 112 Paper III 120 Paper III: Supplementary Information 127 Paper IV 137 Paper V 155 Chapter 1 Introduction Robert Millikan performed the first measurement of the elementary electronic charge, e, in 1909 in his famous oil drop experiment and showed that electrical charge is quantized [1]. He received the Nobel prize in physics 14 years later for that but it took almost half a century before the single electron effects were observed in granu- lar films [2, 3, 4, 5, 6]. In the 1980’s the first electrical circuits which could deal with individual electrons were demonstrated [7]. Today, the electrical systems in which the discrete nature of the electrons is studied are known as the single electronics [8, 9, 10]. The working principle of these circuits are based on quantum tunneling through a very thin insulating layer sandwiched between two conductors, a structure which is known as a tunnel junction [11, 12]. Consider a very small metallic granule which is isolated from the environment by a tunnel junction. Electrons can only move in and out of the granule by tunnel- ing through this junction. If the granule is uncharged in the beginning and then an electron tunnels in, it charges the granule with negative charge, e. Now, if another electron wants to tunnel in, it should overcome the repelling Coulomb− force between the negative electron charge and the negatively charged granule. If the granule is small enough, this repelling Coulomb force can be large so that, for the right cir- cumstances, it prevents the extra electron to tunnel in. This effect is known as the Coulomb blockade and allows us to isolate exactly one extra electron on the metallic granule. If one compares electrical current, the flow of electrons through a con- ductor, to the water flowing in a pipe, then the tunnel junction acts as a dripping tap, letting the electrons to pass (tunnel) one at a time [9]. Today, exploiting the Coulomb blockade, more complex circuits and devices have been developed which allow us to control and manipulate the individual electrons. Single electron devices, due to their working principle, are very charge sensitive. They detect charges as small as a fraction of an electron charge. This unique charge sensitivity, has led to several potential applications for these devices as electrom- eters. In practice, however, there is a problem to exploit these devices as reliable electrometers. They not only detect the charge changes in the system under study, 1 2 Introduction but also any other charge motion or charge reconfiguration in their vicinity will af- fect their output. From the early days of single electronics, many efforts have been made to improve the fabrication and measurement techniques in order to overcome this problem and isolate these devices from the external noise sources [13]. Even when one uses advanced measurement techniques and instruments to elim- inate the external noise [14], there are still intrinsic charge movements at the atomic and molecular levels which disturb the performance of these devices. One of this noise processes is the low frequency charge noise with a power spectrum close to 1=f, where f is the frequency.